# The spectral theorem and direct integrals

I'm wondering if there are any good references that discusses the spectral theorem in terms of direct integrals? I suppose the statement would be something like this:

Let $N \in \mathcal{B}(H)$ be a normal operator. Then there exists a $\sigma$-finite Borel measure $\mu$ on $\sigma(N)$ and a measurable field of Hilbert spaces $\{ H(z): z \in \sigma(N) \}$ such that $H$ is isomorphic to $$\int_{\sigma(N)}^\oplus H(z) \, d \mu(z)$$ and $N$ is unitarily equivalent to $$\int_{\sigma(N)}^\oplus z \, d \mu(z).$$

Is this correct?

• Yes. see J. Dixmier's book on $C^*$ algebras, or Sakai's (also vonNeumann algebras, formerly called $W^*$ algebras). Some of Dixmier's discussion is a little over-complicated because he's trying to avoid using separability of the Hilbert space. There's also a good book by Murphy on $C^*$ algebras. And A. Robert's book on representations of locally compact groups touches on these things. The non-commutative case is "more interesting", of course, but the "easy" commutative case is already interesting for applications, as you say. – paul garrett Jul 3 '17 at 21:33
• Yes, the statement is correct. However, it's certainly not the version of the spectral theorem you would normally want to use. It has its merits in certain specialized situations, for example periodic Schrodinger operators, and in fact Reed-Simon discuss it in this context. – user138530 Jul 10 '17 at 0:28
• A theorem of this sort for unbounded operators in the setting of rigged Hilbert spaces is the subject of Chapter 15 in Volume 2 of Berezansky–Sheftel–Us as well as of Chapter 1 in Volume 4 of Gelfand et al.. – Alex Shpilkin Mar 29 at 7:24